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Full text of "Treatise On Analysis Vol-Ii"

16 XII TOPOLOGY AND TOPOLOGICAL ALGEBRA Let (Ea)ael be any family of uniformizable spaces, and let E = fj Ea. ael For each ael, let (dat^)^eL<x ^e a family of pseudo-distances defining the topology of Ea. If we put (12.5.1) e«^(x,y) = da^(prxx9 ptay) for each pair (x, y) of elements of E = ]^[ Ea, then it is immediately checked ael that the ea> A are pseudo-distances on E. The product of the topologies of the Ea is then the topology on E defined by the ea^ as a runs through I and, for each a e I, A runs through La. The set E endowed with this topology is called the product of the spaces Ea. When I = {1,2} this definition agrees with that given above for two spaces E1? E2. For every finite family (a£, Ai)1^i^n and every finite family (r^igign °f strictly positive real numbers, the set B(x; ((a£, A£)), (rf)) can be written in the form f| Ba where, if a = af, we have Ba = B(praix; (A,-), (r,-)), where j ael runs through the indices for which a,- = a£, and Ba = Ea if a is not equal to any oct. From this and (3.6.4) we deduce immediately: (12.5.2) For each ael, the projection pra is a continuous mapping 0/E onto Ea, and the image under pra of any open set in E is an open set in Ea. (12.5.3) For each finite subset H of I, and each family (Ua)aeH, -where Ua is open in Ea/0r each a e H, the set I fj Ua\ x / fj Ea\ is open in E. \aeH / \aeI-H / For it is the intersection of the sets pr^UJ, where a e H, and these sets are open (12.5.2) and finite in number. The sets described in (12.5.3) are called elementary sets in E. The descrip- tion given above of the neighborhoods shows that (having regard to (3.9.3)) these sets form a basis for the topology of E. Incidentally, this shows also that the topology of E depends only on the topologies of the Ea and not on the families of pseudo-distances (rfa> A)- Moreover, if Sa is a basis for the topology of Ea, the elementary sets such that Ua e 23a for all a form a basis for the topology of E. It should be noted that, if I is infinite, a product f] Ua with ael Ua open in Ea, nonempty and ^ Ea for all ael, is not open in E. (12.5.4) Let Aa be a subset of Ea, for each a, and let A = f] Aa. Then ael A = £| Aa. In particular, if each Aa is closed in Ea, then A is closed in E. aelo-distances on E, and the definition of neigh-